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If you split a golden rectangle along the diagonal, you get a right triangle with sides of 1 and phi. I’ve been trying to figure out what the angles in such a triangle would be. Obviously, I can plug phi or 1/phi into the arctan function, but I’ve been trying to figure out if there is some clean multiple of pi that expresses those angles.

Any one have thoughts or suggestions on how to solve this?

Liam Hancock
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  • It won't be a rational multiple of $\pi$. You will need Galois theory to prove this. – anon Nov 15 '20 at 00:34

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This is a comment that is too long to be fit in a comment.

This reddit thread: https://www.reddit.com/r/math/comments/9std3s/arctangent_of_two_and_the_golden_ratio/ contains a beautiful Proof Without Words of the result:

$$\arctan\left(\frac1\phi\right)=\frac {\arctan 2}2$$

By the result in [Is] ArcTan(2) a rational multiple of $\pi$? (where the answers actually extends the results and determine exactly when the arctan of a rational number is a rational multiple of $\pi$), we see that $\arctan 2$ is indeed NOT a rational multiple of $\pi$. Hence it follows that neither $\arctan (\phi)$ nor $\arctan (\phi^{-1})$ could be rational multiples of $\pi$. The may not be a full answer though, since "clean multiple" is subjective.

player3236
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